1,049 research outputs found
Parameterized (in)approximability of subset problems
We discuss approximability and inapproximability in FPT-time for a large
class of subset problems where a feasible solution is a subset of the input
data and the value of is . The class handled encompasses many
well-known graph, set, or satisfiability problems such as Dominating Set,
Vertex Cover, Set Cover, Independent Set, Feedback Vertex Set, etc. In a first
time, we introduce the notion of intersective approximability that generalizes
the one of safe approximability and show strong parameterized inapproximability
results for many of the subset problems handled. Then, we study approximability
of these problems with respect to the dual parameter where is the
size of the instance and the standard parameter. More precisely, we show
that under such a parameterization, many of these problems, while
W[]-hard, admit parameterized approximation schemata.Comment: 7 page
Parametric and nonparametric symmetries in graphical models for extremes
Colored graphical models provide a parsimonious approach to modeling
high-dimensional data by exploiting symmetries in the model parameters. In this
work, we introduce the notion of coloring for extremal graphical models on
multivariate Pareto distributions, a natural class of limiting distributions
for threshold exceedances. Thanks to a stability property of the multivariate
Pareto distributions, colored extremal tree models can be defined fully
nonparametrically. For more general graphs, the parametric family of
H\"usler--Reiss distributions allows for two alternative approaches to colored
graphical models. We study both model classes and introduce statistical
methodology for parameter estimation. It turns out that for H\"usler--Reiss
tree models the different definitions of colored graphical models coincide. In
addition, we show a general parametric description of extremal conditional
independence statements for H\"usler--Reiss distributions. Finally, we
demonstrate that our methodology outperforms existing approaches on a real data
set.Comment: 24 pages, 9 figure
Parameterizing by the Number of Numbers
The usefulness of parameterized algorithmics has often depended on what
Niedermeier has called, "the art of problem parameterization". In this paper we
introduce and explore a novel but general form of parameterization: the number
of numbers. Several classic numerical problems, such as Subset Sum, Partition,
3-Partition, Numerical 3-Dimensional Matching, and Numerical Matching with
Target Sums, have multisets of integers as input. We initiate the study of
parameterizing these problems by the number of distinct integers in the input.
We rely on an FPT result for ILPF to show that all the above-mentioned problems
are fixed-parameter tractable when parameterized in this way. In various
applied settings, problem inputs often consist in part of multisets of integers
or multisets of weighted objects (such as edges in a graph, or jobs to be
scheduled). Such number-of-numbers parameterized problems often reduce to
subproblems about transition systems of various kinds, parameterized by the
size of the system description. We consider several core problems of this kind
relevant to number-of-numbers parameterization. Our main hardness result
considers the problem: given a non-deterministic Mealy machine M (a finite
state automaton outputting a letter on each transition), an input word x, and a
census requirement c for the output word specifying how many times each letter
of the output alphabet should be written, decide whether there exists a
computation of M reading x that outputs a word y that meets the requirement c.
We show that this problem is hard for W[1]. If the question is whether there
exists an input word x such that a computation of M on x outputs a word that
meets c, the problem becomes fixed-parameter tractable
Exact bosonization of the Ising model
We present exact combinatorial versions of bosonization identities, which
equate the product of two Ising correlators with a free field (bosonic)
correlator. The role of the discrete free field is played by the height
function of an associated bipartite dimer model. Some applications to the
asymptotic analysis of Ising correlators are discussed.Comment: 35 page
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